Simplify and expand the following expression: $ \dfrac{3x + 5}{x - 2}-\dfrac{x}{x + 5} $
Answer: In order to subtract expressions, they must have a common denominator. Get both fractions over a common denominator of $(x - 2)(x + 5)$ Multiply the first term by $\dfrac{x + 5}{x + 5}$ $ \begin{align*} \dfrac{3x + 5}{x - 2} \times \dfrac{x + 5}{x + 5} & = \dfrac{(3x + 5)(x + 5)}{(x - 2)(x + 5)} \\ & = \dfrac{3x^2 + 20x + 25}{(x - 2)(x + 5)}\end{align*} $ Multiply the second term by $\dfrac{x - 2}{x - 2}$ $ \begin{align*} \dfrac{x}{x + 5} \times \dfrac{x - 2}{x - 2} & = \dfrac{(x)(x - 2)}{(x + 5)(x - 2)} \\ & = \dfrac{x^2 - 2x}{(x + 5)(x - 2)}\end{align*} $ Now we have: $ = \dfrac{3x^2 + 20x + 25}{(x - 2)(x + 5)} - \dfrac{x^2 - 2x}{(x + 5)(x - 2)} $ Now both terms have a common denominator we can subtract the numerators: $ = \dfrac{3x^2 + 20x + 25 - (x^2 - 2x)}{(x - 2)(x + 5)} $ $ = \dfrac{3x^2 + 20x + 25 - x^2 + 2x}{(x - 2)(x + 5)} $ $ = \dfrac{2x^2 + 22x + 25}{(x - 2)(x + 5)}$ Expand the denominator: $ = \dfrac{2x^2 + 22x + 25}{x^2 + 3x - 10}$